The Stefan Problem of a Polymorphous Material

1979 ◽  
Vol 46 (4) ◽  
pp. 789-794 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Power Series ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Similarity Solution ◽  
Existence And Uniqueness ◽  
Series Solutions ◽  
Special Cases ◽  
Interfacial Boundary ◽  
Infinite Region ◽  
Uniformly Convergent

The problem of freezing or melting of a polymorphous material in a semi-infinite region with arbitrarily prescribed initial and boundary conditions is studied. Exact solutions of the problem are established. The solutions of temperature of all phases are expressed in polynomials and functions in the error integral family and time t and the position of the interfacial boundaries in power series of t1/2. Existence and uniqueness of the series solutions are considered and proved. It is also shown that these series are absolutely and uniformly convergent. The paper concludes with some remarks on density changes at the interfacial boundary and various special cases, one of which is the similarity solution.

The Stefan Problem With an Imperfect Thermal Contact at the Interface

1982 ◽  
Vol 49 (4) ◽  
pp. 715-720 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Exact Solution ◽  
Stefan Problem ◽  
Existence And Uniqueness ◽  
Mixed Type ◽  
Thermal Contact ◽  
Mathematical Properties ◽  
Interfacial Boundary ◽  
Infinite Region ◽  
Uniformly Convergent ◽  
Basic Solutions

The Stefan problem in a semi-infinite region with arbitrarily prescribed initial and boundary conditions, subject to a condition of the mixed type at the interface is investigated. To establish the exact solution of the problem, some new basic solutions of the heat equation are offered. Their mathematical properties are also supplied. The exact solutions of the temperatures in both phases and of the interfacial boundary are derived in infinite series. The existence and uniqueness of these series are considered and proved. It is also shown that these series are absolutely and uniformly convergent. Some concluding remarks about the differences between the present problem and the classical Stefan problem are given. Also the effect of a density discontinuity at the interface is discussed.

Free In-Plane Vibration Analysis of Annular Plates with General Boundary Conditions

2013 ◽  
Vol 572 ◽  
pp. 189-192
Author(s):  
Dong Yan Shi ◽  
Xian Jie Shi ◽  
Wen L. Li ◽  
Zheng Rong Qin
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Current Method ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Series Solutions ◽  
Plane Vibration ◽  
Annular Plates ◽  
Special Cases

An analytical method is derived for the free in-plane vibration analysis of annular plates with general boundary conditions. Under this framework, all the classical homogeneous boundary conditions can be treated as the special cases when the stiffness for each restraining springs is equal to either zero or infinity. The improved Fourier series solutions for the in-plane vibrations are obtained by employing the Rayleigh-Ritz method. A numerical example is presented to demonstrate the accuracy and reliability of the current method.

scholarly journals Exact Solutions and Conservation Laws of the Time-Fractional Gardner Equation with Time-Dependent Coefficients

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2434
Author(s):  
Ruixin Li ◽  
Lianzhong Li
Keyword(s):  
Power Series ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Symmetry ◽  
Time Dependent ◽  
Series Solutions ◽  
Lie Symmetry Analysis ◽  
Gardner Equation ◽  
Power Series Solutions ◽  
Conservation Theorem

In this paper, we employ the certain theory of Lie symmetry analysis to discuss the time-fractional Gardner equation with time-dependent coefficients. The Lie point symmetry is applied to realize the symmetry reduction of the equation, and then the power series solutions in some specific cases are obtained. By virtue of the fractional conservation theorem, the conservation laws are constructed.

A Method for Exact Series Solutions in Structural Mechanics

1999 ◽  
Vol 66 (2) ◽  
pp. 380-387 ◽  
Author(s):  
J. T.-S. Wang ◽  
C.-C. Lin
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Integral Form ◽  
Structural Mechanics ◽  
Series Solutions ◽  
Analysis Method ◽  
Weighting Functions ◽  
Systematic Analysis ◽  
Unified Method

A systematic analysis method for solving boundary value problems in structural mechanics is presented. Euler-Lagrange differential equations are transformed into integral form with respect to sinusoidal weighting functions. General solutions are represented by complete sets of functions without being concerned with boundary conditions in advance while all boundary conditions are satisfied in the process. The convergence of results is assured, and the procedure leads to pointwise exact solutions. A number of simple structural mechanics problems of stress, buckling, and vibration analyses are presented for illustrative purposes. All results have verified the exactness of solutions, and indicate that this unified method is simple to use and effective.

The Exact Solution to an Ablation Problem With Arbitrary Initial and Boundary Conditions

1984 ◽  
Vol 51 (4) ◽  
pp. 821-826 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Surface Temperature ◽  
Exact Solutions ◽  
Melting Temperature ◽  
Physical Interpretation ◽  
Existence And Uniqueness ◽  
Frictional Heating ◽  
Convective Motions ◽  
Initial And Boundary Conditions

The problem of ablation by frictional heating in a semi-infinite solid with arbitrarily prescribed initial and boundary conditions is investigated. The study includes all convective motions caused by the density differences of various phases of the materials. It is found that there are two cases: (i) ablation appears immediately and (ii) there is a waiting period of redistribution prior to ablation. The exact solutions of velocities and temperatures of both cases are derived. The solutions of the interfacial positions are also established. Existence and uniqueness of the solutions are examined and proved. The conditions for the occurrence of these two cases are expressed by an inequality. Physical interpretation of the inequality is explored. Its implication coincides with one’s expectation. Ablation appears only when the surface temperature is at or above the melting temperature.

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

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